[proofplan] We separate two kinds of information. The front-door formula identifies total-effect interventional distributions using single-world intervention quantities, while natural mediation effects require a joint coupling between counterfactuals from incompatible intervention worlds. We construct two binary counterfactual couplings with the same single-intervention marginal laws and the same total-effect laws under both treatment levels, but with different values of the nested counterfactual $\mathbb E[Y_{1,M_0}]$. This proves that the front-door-identified information alone cannot determine the natural mediation component. [/proofplan]

[step:Fix the single-intervention marginal information shared by both couplings]Let \begin{align*} \Omega:=\{0,1\}^3 \end{align*} and let $\mathcal F:=2^\Omega$. Define $\mathbb P:\mathcal F\to[0,1]$ to be the uniform probability measure on $\Omega$. Let \begin{align*} W,V,R:(\Omega,\mathcal F)\to(\{0,1\},2^{\{0,1\}}) \end{align*} be the coordinate maps. Then $W,V,R$ are independent binary random variables satisfying \begin{align*} \mathbb P(W=1)=\mathbb P(V=1)=\mathbb P(R=1)=\frac12. \end{align*} In both counterfactual couplings define mediator potential outcomes \begin{align*} M_0:=W \end{align*} and \begin{align*} M_1:=V. \end{align*} Thus \begin{align*} \mathbb P(M_0=1)=\mathbb P(M_1=1)=\frac12. \end{align*} We now define two joint laws for the outcome counterfactuals $Y_{a,m}$. For the first coupling, define \begin{align*} Y_{0,0}:=R,\quad Y_{0,1}:=R,\quad Y_{1,0}:=R,\quad Y_{1,1}:=R. \end{align*} For the second coupling, define \begin{align*} Y_{0,0}:=R,\quad Y_{0,1}:=R,\quad Y_{1,0}:=1-W,\quad Y_{1,1}:=W. \end{align*} In both couplings, for every $a,m\in\{0,1\}$, \begin{align*} \mathbb P(Y_{a,m}=1)=\frac12. \end{align*} Indeed, this is immediate for the variables equal to $R$, and in the second coupling it also holds for $Y_{1,0}=1-W$ and $Y_{1,1}=W$ because $W$ has the Bernoulli law with parameter $1/2$. Therefore the two couplings agree on the marginal laws of every $M_a$ and every $Y_{a,m}$.[/step]

[guided]The point of the construction is to keep fixed the quantities that single-intervention front-door calculations can see, while changing how variables from different intervention worlds are coupled. We use the finite [probability space](/page/Probability%20Space) \begin{align*} \Omega:=\{0,1\}^3 \end{align*} with $\mathcal F:=2^\Omega$ and uniform probability measure $\mathbb P$. The coordinate maps \begin{align*} W,V,R:(\Omega,\mathcal F)\to(\{0,1\},2^{\{0,1\}}) \end{align*} are independent binary random variables, each taking the value $1$ with probability $1/2$ and the value $0$ with probability $1/2$. In both couplings we define the mediator potential outcomes by \begin{align*} M_0:=W \end{align*} and \begin{align*} M_1:=V. \end{align*} Thus the marginal mediator laws are fixed: \begin{align*} \mathbb P(M_0=1)=\mathbb P(W=1)=\frac12 \end{align*} and \begin{align*} \mathbb P(M_1=1)=\mathbb P(V=1)=\frac12. \end{align*} For the first coupling, every outcome-response variable is the same Bernoulli variable $R$: \begin{align*} Y_{0,0}:=R,\quad Y_{0,1}:=R,\quad Y_{1,0}:=R,\quad Y_{1,1}:=R. \end{align*} For the second coupling, the untreated-outcome response variables remain equal to $R$, but the treated-outcome response variables are tied to $W$: \begin{align*} Y_{0,0}:=R,\quad Y_{0,1}:=R,\quad Y_{1,0}:=1-W,\quad Y_{1,1}:=W. \end{align*} Every one of these outcome-response variables takes the value $1$ with probability $1/2$. For $R$ this follows from $\mathbb P(R=1)=1/2$. For $W$ it follows from $\mathbb P(W=1)=1/2$. For $1-W$, we compute \begin{align*} \mathbb P(1-W=1)=\mathbb P(W=0)=\frac12. \end{align*} Hence, for every $a,m\in\{0,1\}$, both couplings satisfy \begin{align*} \mathbb P(Y_{a,m}=1)=\frac12. \end{align*} So the single-intervention marginal information is identical in the two constructions. The only thing we have changed is the cross-world dependence structure.[/guided]

[step:Verify that the total-effect laws are unchanged] For treatment $0$, both couplings satisfy $Y_{0,0}=Y_{0,1}=R$. Since $M_0$ is $\{0,1\}$-valued, both couplings have \begin{align*} Y_{0,M_0}=R. \end{align*} Thus the total-effect law of $Y_{0,M_0}$ is the law of $R$ in both couplings. In the first coupling, \begin{align*} Y_{1,M_1}=R \end{align*} because $Y_{1,0}=Y_{1,1}=R$. Therefore \begin{align*} \mathbb E[Y_{1,M_1}]=\mathbb E[R]=\frac12. \end{align*} In the second coupling, since $M_1=V$, $Y_{1,0}=1-W$, and $Y_{1,1}=W$, we have \begin{align*} Y_{1,M_1}=(1-V)(1-W)+VW. \end{align*} Using independence of $V$ and $W$, \begin{align*} \mathbb E[Y_{1,M_1}]=\mathbb E[(1-V)(1-W)]+\mathbb E[VW]. \end{align*} The two terms are \begin{align*} \mathbb E[(1-V)(1-W)]=\mathbb P(V=0,W=0)=\frac14 \end{align*} and \begin{align*} \mathbb E[VW]=\mathbb P(V=1,W=1)=\frac14. \end{align*} Hence \begin{align*} \mathbb E[Y_{1,M_1}]=\frac12. \end{align*} Thus the two couplings agree on the total-effect law of $Y_{1,M_1}$, because $Y_{1,M_1}$ is binary and its law is determined by the common expectation $\mathbb E[Y_{1,M_1}]=\frac12$. Therefore they agree on the total-effect information for both treatment levels. [/step]

[step:Compute the cross-world nested counterfactual under the mediator from treatment $0$] In the first coupling, $Y_{1,0}=Y_{1,1}=R$, so \begin{align*} Y_{1,M_0}=R. \end{align*} Therefore \begin{align*} \mathbb E[Y_{1,M_0}]=\frac12. \end{align*} In the second coupling, $M_0=W$, $Y_{1,0}=1-W$, and $Y_{1,1}=W$. Hence \begin{align*} Y_{1,M_0}=(1-W)Y_{1,0}+WY_{1,1}. \end{align*} Substituting the definitions of $Y_{1,0}$ and $Y_{1,1}$ gives \begin{align*} Y_{1,M_0}=(1-W)(1-W)+W^2. \end{align*} Since $W$ is $\{0,1\}$-valued, $(1-W)^2=1-W$ and $W^2=W$. Thus \begin{align*} Y_{1,M_0}=1. \end{align*} Consequently \begin{align*} \mathbb E[Y_{1,M_0}]=1. \end{align*} [/step]

[step:Conclude that front-door identification alone does not identify natural mediation effects] The two couplings agree on the marginal laws of all single-intervention variables $M_a$ and $Y_{a,m}$, and they agree on the total-effect laws of $Y_{0,M_0}$ and $Y_{1,M_1}$. These are the kinds of single-world quantities determined by a front-door total-effect identification. However, they disagree on the nested cross-world counterfactual: \begin{align*} \mathbb E[Y_{1,M_0}]=\frac12 \end{align*} in the first coupling, while \begin{align*} \mathbb E[Y_{1,M_0}]=1 \end{align*} in the second coupling. Therefore no functional depending only on the front-door-identified single-intervention information can determine $\mathbb E[Y_{1,M_0}]$. Natural direct and indirect effects require additional assumptions specifying the joint distribution of cross-world potential outcomes, not merely identification of total-effect interventional laws. This proves the claim. [/step]